# 3 - Derivative of a function

## Theory

The derivative enables us to calculate the slope of a graph. Take a function f and a point a on the X-axis. We are going to determine the slope of the graph at the point M, whose abscissa is a.

Its slope is equal to the slope of its tangent at the same point. You know how to calculate the slope of a line that goes through two points A and B thanks to the formula

But here there is only one point, M. Take a number h and place the point N, whose abscissa will be a+h.

The coordinates of M and N are :

Hence the slope of the line (MN) is :

We remark that the smaller h gets, the closer to the red line the green line gets, that means that when h tends to 0, the number gets closer to the slope of the red line when x = a. The number, when it exists, is equal to the slope of the graph when x = a. It's the number we were looking for.
We denote it . It is the derivative of f at a.

On the drawing below, is positive because the tangent goes up that means the slope of the tangent is positive. is positive too but is smaller because the tangent is less steep. We also have and

## How to calculate the derivative at a point

How to calculate the derivative of the function at point x = 2 :

We have seen on the graph what the derivative of a function at a point is, it's the slope of the tangent to the graph at this point. You've seen how to calculate it (right up above). For every function f, you can associate a number f'(a) to each number a, that enables us to define the derivative function. The derivative of a function is the function to which you give a number and gives back the number f'(a). Below, on the left the example of a function f and on the right its derivative f'.

## How to calculate the derivative of a function

Now you're going to learn the rules to calculate the derivative of a function without doing such a hard calculation as above. You have to know the derivatives of the main functions and the derivation rules. The derivative is written under each example.
The derivative of a constant function is always 0 (the tangent is horizontal, hence its slope is 0).

## Derivation rules

The derivative of a sum of functions is equal to the sum of the derivatives. For the product, the quotient, or the composition (function of fonction), it's not that simple. If u and v are two functions, then :

You need to know all these formulas by heart.

## How to use the formulas

Indeed, if k is a number, the derivative of ku is always ku'.

*** Calculate the derivative of . Set and . Hence and .
Then: .

*** Calculate the derivative of .
Set and . Then and . Hence :

You can expand then reduce the numerator but don't touch the denominator (useful for next chapter).

*** Calculate the derivative of .
Set and , then : and . Hence :

Don't forget the brackets in the expression before the sign = , and congratulations you've read the whole page !

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Derivative of a function

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